# Why don't parabolas have asymptotes?

an asymptote is a line which becomes the tangent of the curve as the $$x$$ or $$y$$ cordinates of the curve tends to infinity.

Hyperbola has asymptotes but parabolas ( both being an open curve and a conic section) do not.

I am just curious to know why don't parabolas have an asymptote ? Is there any mathematical proof to show that ?

• How do you define a parabola? Commented Sep 5, 2020 at 5:24
• @Andrew Chin it is the locus of points whose distance from a fixed line and from a fixed point is equal. Commented Sep 5, 2020 at 5:40
• Add a link to the page Commented Sep 5, 2020 at 5:56
• Hyperbolas are the only conic sections with asymptotes. Even though parabolas and hyperbolas look very similar, parabolas are formed by the distance from a point and the distance to a line being the same. Therefore, parabolas don’t have asymptotes. Commented Sep 5, 2020 at 6:58
• @SarGe this is the line which I couldn't understand. Can you make it clear ? Commented Sep 5, 2020 at 7:00

If a parabola had an asymptote, then we could translate-rotate it so that the asymptote coincides with the $$y$$ axis.

Now, consider the general conic section (which, of course, includes all rotated parabolas):

$$A x^2 + B x y + C y^2 + D x + E y + F= 0 \tag1$$ Or $$y( C y + B x + E) + A x^2 + D x + F=0 \tag2$$

and let's see under what conditions it could happen its graphic has the $$y$$ axis as an asymptote, i.e. that as $$x\to 0$$ $$y \to \pm\infty$$.

In that limit, we must have $$( C y + B x + E) \to -\frac{F}{y} \to 0\tag3$$

But this requires $$C=0$$ and $$E=0$$

Then it cannot be a parabola (or an ellipse), only an hyperbola.

• Great answer! But for $( C y + B x + E) \to 0$, isn't $y \to \frac{-E}{C}$ sufficient? And for that, the only conclusions I can see are $E \to \pm \infty$, $C \to 0$ or both. How is it only $C=E=0$ that you got as a solution? Admittedly, I've never heard anyone else talk about coefficients being limits, but this is what I can conclude analytically. Where did I go wrong? Commented Aug 26, 2021 at 15:40
• @harry When want both $x\to 0$ and $y \to \infty$ to happen. Commented Aug 26, 2021 at 17:20
• But isn't that the condition that's required? That's what you've used in your answer, right? Commented Aug 26, 2021 at 18:17
• Since arms of a parabola become parallel as distance increases. Shouldn't the two parallel lines that they are approaching/going to become considered the asymptotes of the parabola? Commented May 14 at 18:43

Hints:

An $$\textit{asymptote}$$ is a straight line to which the curve approaches while it moves away from the origin. The curve can approach the line from one side, or it can intersect it again and again. Not every curve which goes infinitely far from the origin (infinite branch of the curve) has an asymptote.

The graph of the function $$y=ax^n$$

where $$\space n>0, \space$$ integer, is a $$\textit{parabola of n-th degree, or of n-th order}.$$

For functions given in explicit form $$\space y=f(x), \space$$ we know: the vertical asymptotes are at points of discontinuity where the function $$\space f(x)\space$$ has an infinite jump; the horizontal and oblique asymptotes have the equation:

$$y=kx+b, \space \text{ with } \space \space k=\lim_{x \to \infty}\frac{f(x)}{x}, \space \space b=\lim_{x \to \infty}\left[f(x)-kx \right].$$